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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习3.4 - 解析函数与调和函数的关系 }
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\date{2024 年 4 月 29 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设二元实变函数 $H(x,y)$ 在区域 $D$ 内有定义。什么时候称 $H(x,y)$ 是区域 $D$ 内的调和函数？

\vspace{0.2cm}

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\item  %Problem 02
什么时候调和函数 $v(x,y)$ 称为是调和函数 $u(x,y)$ 的共轭调和函数？

\vspace{0.2cm}

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\item  %Problem 03
设复变函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内解析。则在区域 $D$ 内 $v(x,y)$ 是 $u(x,y)$ 的共轭调和函数。

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\item  %Problem 04
设 $u(x,y)$ 在单连通区域 $D$ 内是调和函数。证明存在区域 $D$ 内的函数 $v(x,y)$, 使得 
$f(z)=u(x,y)+iv(x,y)$ 是 $D$ 内的调和函数。

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\item  %Problem 05
验证 $u(x,y)=x^3-3xy^2$ 是 $z$ 平面上的调和函数，并求解析函数 $f(z)=u(x,y)+iv(x,y)$ 使得 $f(0)=i$. 

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\item  %Problem 06
验证 $v(x,y)=\arctan\frac{y}{x}$ 是右半平面 $D=\{(x,y)\mid x>0\}$ 上是调和函数，并求以 $v(x,y)$ 为虚部的解析函数。

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\item  %Problem 07
求解析函数 $f(z)=u+iv$, 使得 $u=x^2+xy-y^2$ 以及 $f(i)=-1+i$. 

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\item  %Problem 08
设函数 $f(z)$ 在区域 $D$ 内解析，证明 
$$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 4 \frac{\partial^2 f(z)}{\partial z \partial \bar{z}}. $$


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%\item  %Problem 09
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%\item  %Problem 10
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\end{enumerate}


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\end{document}

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